I've been given the parabolic equation:
$ 3 \frac{∂^2u}{∂x^2} + 6\frac{∂^2u}{∂x∂y} + 3 \frac{∂^2u}{∂y^2} - \frac{∂u}{∂x} - 4\frac{∂u}{∂y} + u = 0$
The questions ask you to find the characteristic coordinates and then put into canonical form. Which I have done. The canonical form being:
$ \frac{∂^2u}{∂η^2} = \frac{∂u}{∂ξ} + \frac{1}{3}\frac{∂u}{∂η} - \frac{1}{3}u $
However I can't work through the last part of the question which says:
Use the substitution $u(ξ, η) = e^{λξ + µη}v(ξ, η)$ and find the constants $λ$ and $µ$ such that the equation can be further reduced to:
$ \frac{∂v}{∂ξ} = v\frac{∂^2v}{∂η^2}$
for some constant $v$ which you also need to identify.
Thanks for any help.